Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution

作者：Moehle M^*; Pitters H

来源：Markov Processes and Related Fields, 2015, 21(2): 317-338.

• absorption time
• cumulants
• Euler-Mascheroni integrals
• Gumbel distribution
• infinite convolution
• Kingman coalescent
• moments
• most recent common ancestor
• tree length
• zeta function

Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T are linear combinations of the values zeta(2m), m is an element of {0, ... , left perpendicularj/2right perpendicular}, of the Riemann zeta function zeta with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length L-n of the Kingman coalescent restricted to a sample of size n is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.

• 出版日期2015